Stability Criteria for Second-Order Linear Differential Equations

When you have eliminated the impossible, whatever remains, however improbable, must be the truth, Sherlock Holmes.

Stability Criteria for Second-Order Linear Differential Equations with Constant Coefficients

We are exploring the stability of solutions to second-order linear differential equation with constant coefficients which take the standard form:of the form: $y'' + Ay' + By = f(t)$ where:

• A and B are constant.
• $y(t)$ is the unknown function of time.
• $y'$ and $y''$ are the first and second derivatives of y with respect to t, respectively
• $f(t)$ is the forcing term, representing an external input or driving force.

Understanding the stability of such equations is crucial in various fields like mechanical engineering, electrical engineering, and physics, as it helps predict the long-term behavior of dynamic systems, determining whether they will settle into a steady state, exhibit sustained oscillations, or experience unbounded growth.

General Solution of the Differential Equation

The general solution to an inhomogeneous differential equation consists of two distinct components:

1. Particular Solution $y_p$. A specific solution that solves the inhomogeneous equation $y'' + Ay' + By = f(t)$. This solution accounts for the system’s direct response to the external forcing term f(t). It does not contain arbitrary constants.
2. Complementary solution $y_c$ (or homogeneous solution). The general solution to the corresponding or associated homogeneous equation $y'' + Ay' + By = 0$. This solution represents the system’s natural behavior without external input. It contains arbitrary constants determined by the system’s initial conditions.

Therefore, the general solution is the sum of both parts: $y(t) = y_p(t) + y_c(t) = y_p(t) + c_1y_1(t) + c_2y_2(t)$ where:

• $y_p$ addresses the external forcing term f(t).
• $y_c(t) = c_1y_1(t) + c_2y_2(t)$ are solutions to the homogeneous equation. It represents the system’s natural, unforced behavior.
• $c_1$ and $c_2$ are arbitrary constants defined by the initial state of the system (like initial position and initial velocity).

Complementary Solution and Stability

To analyze system stability, we focus entirely on the complementary solution $y_c(t)$ because it governs how the system behaves over time independent of ongoing external forces.

Solving the Homogeneous Equation

To find the complementary solution $y_c(t)$, we solve the homogeneous equation: $y'' + Ay' + By = 0$.

Assuming a solution of the form $y(t) = e^{rt}$, where r is a constant, we substitute this into the homogeneous equation and gives us the characteristic equation: $r^2e^{rt}+ Are^{rt} + Be^{rt} = 0$.

Dividing by $e^{rt}$ (since an exponential function is never zero), we get the characteristic equation: $r^2 + Ar + B = 0$

Form of the Complementary Solution

The roots of this quadratic equation ($r_1$ and $r_2$) determine the form of the complementary solution. The behavior of the complementary solution and thus, the stability of the ODE depends on these roots.

Let’s examine the different types of characteristic roots and their corresponding solutions:

1. Distinct Real Roots ($r_1 \neq r_2$):$y_c(t) = c_1e^{r_1t} + c_2e^{r_2t}$. The solution is stable if both $r_1 < 0$ and $r_2 < 0$

2. Repeated Real Roots ($r = r_1 = r_2$): $y_c(t) = (c_1 + c_2t)e^{rt}$. The solution is stable if r < 0.

3. Complex Roots ($r = \alpha \pm \beta i$): $y_c(t) = ^{\alpha t}(c_1\cos(\beta t) + c_2\sin(\beta t))$. The solution is stable if $\alpha < 0$.

Where:

• $c_1$ and $c_2$ are constants determined by initial conditions.
• $\alpha = \text{Re}(r)$ is the real part of the complex root.
• $\beta = \text{Im}(r)$ is the imaginary part of the complex root.

Defining Stability

In this context, stability refers to the behavior of the complementary solution $y_c(t)$ as time t approaches infinity ($t \to \infty$). Specifically, we want to know if the expression $c_1y_1 + c_2y_2$ vanishes over time for any initial conditions.

• Stable Solution. If $y_c(t) \to 0$ as $t \to \infty$ for all initial conditions (i.e., for all values of $c_1$ and $c_2$), the solution or the system is said to be stable. The transient part of the solution, $y_c(t)$, decays over time and the system settle into the steady-state behaviour described by the particular solution $y_p$. Inicial disturbances or conditions have diminishing effects as time progresses, they basically “die out”.
• Unstable Solution. If $y_c(t)$ does not approach zero as $t \to \infty$, meaning it grows without bound or oscillates indefinitely without shrinking, the solution or system is unstable. The transient part of the solution persists or explodes over time, preventing the system from ever reaching a steady state.Initial conditions or disturbances have lasting or increasing effects.

Core Rule of Thumb: The second-order system $y'' + Ay' + By = f(t)$ is stable if and only if all characteristic roots have negative real parts.

Analysis of Stability Based on Characteristic Roots

1. Distinct real roots $r_1$ and $r_2$. The complementary solution for distinct real roots is: $y_c(t) = C_1e^{r_1t}+C_2e^{r_2t}$.

   For the solution to decay exponentially to zero as $t \to \infty$, both exponential terms must tend to zero. This occurs if both roots $r_1$ and $r_2$ are negative. Therefore, the ODE is stable if both characteristic roots are negative.

   If either root $r_1$ or $r_2$ is positive, the corresponding exponential term will grow exponentially, leading to an unstable solution.

2. Repeated real root $r_1 = r_2 = t$. When the characteristic equation has a repeated real root $r_1 = r_2 = t$, the complementary solution takes the form: $y_c(t) = (C_1+C_2t)e^{rt}$.

   In this case, the solution is stable if r < 0 because $e^{rt} \to 0$ (decays exponentially). The polynomial term ($C_1+C_2t$) may grow with t, but since it’s multiplied by a decaying exponential, the overall solution still approaches zero.

   If r = 0, the solution becomes $y_c(t) = (C_1+C_2t)$, which does not decay to zero unless $C_1 = C_2 = 0$, so it’s unstable, too.

   If r > 0, the exponential term grows exponentially, leading to instability.

3. Complex Roots $r = \alpha \pm \beta i$. When the characteristic roots are complex, the complementary solution is of the form: $e^{\alpha t}(c_1\cos(\beta t) + c_2\sin(\beta t))$. Here, the oscillatory terms cos(bt) and sin(bt) are multiplied by the exponential factor $e^{\alpha t}$. The exponential term dictates the amplitude of the oscillations.

   The behavior of the solution depends on the real part $\alpha$. If $\alpha \lt 0$, the exponential term $e^{\alpha t}$ decays, causing the oscillations to diminish over time, and the solution is stable.

   The oscillatory terms cos(βt) and sin(βt) are bounded between −1 and 1, so the exponential term dictates the overall behavior.

   If $\alpha = 0$, the solution oscillates indefinitely without decaying, the amplitude remains constant, leading to sustained oscillations (this is a case of marginal stability).

   If $\alpha \gt 0$, the exponential term grows, the oscillations increase in amplitude, making the solution unstable.

To summarize, the second-order $y'' + Ay' + By = f(t)$ is stable if the characteristic roots have negative real parts.

Physical Interpretation of Stability

In physical systems, understanding the concept of stability is crucial for predicting how physical systems behave over time. It refers to the system’s response to initial disturbances or external forces and whether the system returns to equilibrium or diverges as time progresses.

Physical systems often exhibit two primary behaviors in response to stimuli:

• Transient behavior. Dictated by the complementary solution. This is how the system reacts or responds initially to initial conditions or disturbances. If the system is stable, this phase is temporary and decays over time.
• Steady-state behavior. Dictated by the particular solution. This is the long-term behavior of the system after the transient effects have faded. It shows how the system synchronizes (or responds) with the continuous external input.

Engineering Examples

• The Spring-Mass-Dashpot System (Mechanical).

Stability means that any initial oscillations or displacements due to an external force or displacement will eventually die out, allowing the system to return to equilibrium. This happens when the damping force, represented by the dashpot (b), is sufficient to counteract the effects of any initial energy input, such as an external force or displacement.

An equation modeling a damped harmonic oscillator (e.g., a mass attached to a spring and a friction-inducing dashpot) is: $my'' + by' + ky = F(t)$ where: m: Mass; b: Damping coefficient (dashpot friction); k: Spring constant; F(t): External driving force.

The characteristic equation is $mr^2 + br + k = 0$. The roots of the characteristic equation, r, determine the system’s behavior. We analyze the discriminant$\Delta = b^2 - 4mk$.

Stability analysis:

1. Overdamped ($b^2 \gt 4mk, \Delta \gt 0$): Two distinct, real, negative roots. Complementary solution: $y_c(t) = c_1e^{r_1t}+c_2e^{r_2t}$. Exponential decay without oscillations. The system returns to equilibrium slowly without oscillating, hence the solution is stable.
2. Critically Damped ($b^2 = 4mk, \Delta = 0$): One repeated real negative root $r=\frac{-b}{2m}$. Complementary solution: $y_c(t)=(c_1+c_2t)e^{rt}$. It is the the fastest to return to equilibrium without oscillations. The system decays to zero as quickly as possible, hence the solution is stable.
3. Underdamped ($b^2 \lt 4mk, \Delta \lt 0$): Complex roots with a negative real part, $\alpha = \frac{-b}{2m} < 0$. Complementary solution: $y_c(t)=e^{\alpha t}(c_1\cos(\beta t) + c_2\sin(\beta t))$. It describes an oscillatory motion with exponentially shrinking amplitude, eventually settling. The system oscillates while gradually returning to equilibrium, hence solution is stable with oscillations.
4. No Damping ($b = 0, \Delta \lt 0$): Purely imaginary roots $r = \pm i \omega_0$. Complementary solution: $y_c(t)=(c_1 \cos(\omega_0 t) + c_2 \sin(\omega_0 t))$. There are sustained oscillations with constant amplitude. The system continues to oscillate indefinitely. It is marginally stable, the system remains bounded but does not return to equilibrium.

• Electrical RLC Circuit

In an electrical circuit, stability implies that initial voltage spikes or current surges will eventually fade, allowing the circuit to reach a safe, steady operating state. The stability of an electrical circuit is influenced by factors such as the nature of the circuit (e.g., resistive, inductive, or capacitive), the level of damping (e.g., due to resistors, inductors, or capacitors), and the operating conditions.

An RLC circuit is governed by: $Lq'' + Rq' + \frac{1}{C}q = E(t)$ where L: Inductance; R: Resistance (acts as electrical damping); C: Capacitance; q(t): Charge; E(t): External voltage source.

Consider the homogeneous equation (E(t) = 0): $Lq'' + Rq' + \frac{1}{C}q = 0$. Assuming a solution $q(t) = e^{rt}$, we substitute into the homogeneous equation: $L(r^2e^{rt}) + R(re^{rt}) + \frac{1}{C}e^{rt} = 0$. Divide both sides by $e^{rt}$. Characteristic Equation: $Lr^2 + Rr + \frac{1}{C} = 0$. Its roots determine the behaviour of the system, we analyze the discriminant: $D = R^2 -\frac{4L}{C}$:

Stability Analysis

1. Overdamped, $R^2 > 4\frac{L}{C}$ (D > 0). Two distinct real negative (R > 0) roots; solution is stable. Complementary solution: $q_c(t) = c_1e^{r_1t}+c_2e^{r_2t}$. There’s an exponential decay of current/charge without oscillations. The circuit returns to steady state slowly, hence the system is stable.
2. Critically Damped, $R^2 = 4\frac{L}{C}$ (D = 0). Repeated real negative root $r=\frac{-R}{2L}$. Complementary solution: $q_c(t)=(c_1+c_2t)e^{rt}$. This system shows the fastest return to steady state without oscillations, hence the solution is stable.
3. Underdamped, $R^2 < 4\frac{L}{C}$ (D < 0). Complex roots (α + βi) with negative real part, $\alpha = \frac{-R}{2L} \lt 0$. Complementary solution: $q_c(t)=e^{\alpha t}(c_1\cos(\beta t) + c_2\sin(\beta t))$. Current oscillates back and forth but diminishes over time. The system oscillates while gradually returning to equilibrium, hence the solution is stable with oscillations.
4. No Resistance (R = 0). Purely imaginary roots. Complementary solution: $q_c(t)=c_1\cos(\beta t) + c_2\sin(\beta t)$. The energy bounces endlessly between the capacitor and the inductor without losing amplitude (Marginally Stable).

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